Search: id:A125581
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%I A125581
%S A125581 77,847,9317,102487,596778,1127357,1193556,6161805,12323610,12400927
%N A125581 Numbers n such that n does not divide the denominator of the n-th harmonic 
               number nor the denominator of the n-th alternating harmonic number.
%C A125581 Note that a(1) = 7*11, a(2) = 7*11^2, a(3) = 7*11^3.
%C A125581 Harmonic numbers are defined as H(n) = Sum[ 1/k, {k,1,n} ] = A001008(n)/
               A002805(n).
%C A125581 Alternating harmonic numbers are defined as H'(n) = Sum[ (-1)^(k+1)*1/
               k, {k,1,n} ] = A058313(n)/ A058312(n).
%C A125581 Numbers n such that n does not divide the denominator of the n-th harmonic 
               number are listed in A074791(n) = {6, 18, 20, 21, 33, 42, 54, 63, 
               66, 77, 100, ...}.
%C A125581 Numbers n such that n does not divide the denominator of the n-th alternating 
               harmonic number are listed in A121594(n) = {15, 28, 75, 77, ...}.
%C A125581 Sequence is the intersection of A074791(n) and A121594(n).
%C A125581 Comments from Max Alekseyev, Mar 07 2007: (Start) While A125581 indeed 
               contains the geometric progression 7*11^n as a subsequence, it also 
               contains other geometric progressions such as: 546*1093^n, 1092*1093^n, 
               1755*3511^n, 3510*3511^n and 4896*5557^n (see A126196 and A126197). 
               It may also contain some "isolated" terms (i.e. not participating 
               in the geometric progressions) but such terms are harder to find 
               and at the moment I have no proof that they exist.
%C A125581 This is a sketch of my proof that geometric progression 7*11^n and the 
               others mentioned above belong to A125581.
%C A125581 Lemma 1. H'(n) = H(n) - H([n/2])
%C A125581 Lemma 2. For prime p and integer n>=p, valuation(H(n),p) >= valuation(H([n/
               p]),p) - 1
%C A125581 Theorem. For an integer b>1 and a prime number p such that p divides 
               the numerators of both H(b) and H([b/2]), the geometric progression 
               b*p^n belongs to A125581.
%C A125581 Proof. It is enough to show that valuation(H(b*p^n),p)>-n and valuation(H'(b*p^n),
               p)>-n. By Lemma 2 we have valuation(H(b*p^n),p) >= valuation(H(b),
               p) - n >= 1-n > -n.
%C A125581 From this inequality Lemma 1 we have valuation(H'(b*p^n),p) >= min{ valuation(H(b*p^n),
               p), valuation(H(b*p^n/2),p) } >= min{ 1-n, valuation(H([b*p^n/2]),
               p) } and we need to show that valuation(H([b*p^n/2]),p) >= 1-n.
%C A125581 Again by Lemma 2, we have valuation(H([b*p^n/2]),p) >= valuation(H([b/
               2]),p) - n >= 1-n that completes the proof.
%C A125581 It is easy to check that this Theorem holds for the aforementioned geometric 
               progressions. (End)
%H A125581 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               NonRecursions.html">Non Recursions</a>
%H A125581 Eric Weisstein, Link to a section of The World of Mathematics. <a href="http:/
               /mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%t A125581 f=0; g=0; Do[g=g+1/n; f=f+(-1)^(n+1)/n; If[ !IntegerQ[Denominator[g]/
               n]&&!IntegerQ[Denominator[f]/n], Print[n]], {n, 1, 10000}]
%Y A125581 Cf. A001008, A002805 = Denominator of the n-th harmonic number. Cf. A058313, 
               A058312 = Denominator of the n-th alternating harmonic number. Cf. 
               A074791 = numbers n such that n does not divide the denominator of 
               the n-th harmonic number. Cf. A121594 = numbers n such that n does 
               not divide the denominator of the n-th alternating harmonic number. 
               Cf. A119955, A003599.
%Y A125581 Sequence in context: A027574 A059047 A046149 this_sequence A093277 A073931 
               A105253
%Y A125581 Adjacent sequences: A125578 A125579 A125580 this_sequence A125582 A125583 
               A125584
%K A125581 hard,more,nonn
%O A125581 1,1
%A A125581 Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 03 2007
%E A125581 More terms from Max Alekseyev, Mar 11 2007
%E A125581 a(8)-a(10) from Max Alekseyev, Mar 19 2007

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